Use the properties of logarithms to find the exact value of the expression log[16]4*log[5]5�ã7
What is 5�ã7 ?
suppose to be SQRT7
is that 5√7 or √7 ?
log164 = 1/2
log5 5√7 is not exact
was that base supposed to be 7 ?
log[16]4*log[5]5�ã7 is what the question, I am unsure how to solve exact
log[16]4*log[5]5sqrt7
log[16]4*log[5]5sqrt7
= (1/2(*(log[5]5 + log[5]√7)
= (1/2)( 1 + (1/2)log[5] 7)
= 1/2 + (1/4) log57
To find the exact value of the expression log[16]4 * log[5]5�ã7, we can use the properties of logarithms.
First, let's simplify log[16]4. Since 16 is a power of 4 (4^2 = 16), we can rewrite log[16]4 as log[4^2]4. According to the property log[b]a^c = c * log[b]a, we can rewrite this expression as 2 * log[4]4.
Now, let's simplify log[5]5�ã7. Since the base of the logarithm is the same as the number inside the logarithm, log[b]b = 1. Therefore, log[5]5�ã7 = 1 * log[5]7.
Now we can substitute these simplified expressions back into the original expression:
log[16]4 * log[5]5�ã7 = (2 * log[4]4) * (log[5]7)
Now, let's evaluate each logarithm individually using base 10 logarithms.
Since log[10]10 = 1, log[4]4 = log[10]10 / log[10]4 = 1 / log[10]4
Similarly, log[5]7 = log[10]7 / log[10]5
Now, we can rewrite the expression with base 10 logarithms:
(2 * (1 / log[10]4)) * (log[10]7 / log[10]5)
Since log[b]a denotes the logarithm of a with base b, we can simplify further:
(2 / log[10]4) * (log[10]7 / log[10]5)
Next, we can simplify the expression by multiplying the numerators and the denominators:
(2 * log[10]7) / (log[10]4 * log[10]5)
Now let's evaluate the logarithms:
log[10]7 ≈ 0.8451
log[10]4 ≈ 0.6021
log[10]5 ≈ 0.6989
Substituting these values into the expression, we get:
(2 * 0.8451) / (0.6021 * 0.6989) ≈ 1.69 / 0.4202 ≈ 4.022
Therefore, the exact value of the expression log[16]4 * log[5]5�ã7 is approximately 4.022.